Abstract
In this work a distributed optimal control problem for time-dependent Burgers equation is analyzed. To solve the nonlinear control problems the augmented Lagrangian-SQP technique is used depending upon a second-order sufficient optimality condition. Numerical test examples are presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
H.W. Alt, Lineare Funktionalanalysis. Eine anwendungsorientierte Einführung, 2nd edn., Springer-Verlag, 1992.
J.M. Burgers, “Application of a model system to illustrate some points of the statistical theory of free turbulence, ” Nederl. Akad. Wefensh. Proc., vol. 43, pp. 2–12, 1940.
J.M. Burgers, “A mathematical model illustrating the theory of turbulence, ” Adv. in Appl. Mech., vol. 1, pp. 171–199, 1948.
J.M. Burgers, “Statistical problems connected with asymptotic solution of one-dimensional nonlinear diffusion equation, ” Statistical Models and Turbulence, pp. 41–60, 1972.
J.A. Burns and S. Kang, “A control problem for burgers' equation with bounded input/output, ” Technical Report 90-45, ICASE Report, June 1990. NASA Contractor Report 182062.
J.A. Burns and S. Kang, “A stabilization problem for Burgers' equation with unbounded control and observation, ” International Series of Num. Math., 100, 1991.
C.I. Byrnes, D.S. Gilliam, and V.I. Shubov, On the global dynamics of a controlled viscous Burgers' equation. preprint, Dec. 17, 1995.
Z. Chen, G. Wang, and S. Weerakoon, “An initial value control problem for the Burgers' equation, ” Distributed Parameter Systems, vol. 75, pp. 52–76, 1985.
H. Choi and R. Temam, “Application of incremental unknowns to the Burgers equation, ” Annual Research Briefs 1992, pp. 129–138, 1992. Center for Turbulence Research.
H. Choi, R. Temam, P. Moin, and J. Kim, “Feedback control for unsteady flow and its application to the stochastic Burgers equation, ” J. Fluid Mech., 1993.
J.D. Cole, “On a quasi-linear parabolic equation occuring in erodynamics, ” Quarterly of Applied Mathematics, vol. IX, no. 3, pp. 225–236, 1951.
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Volume 5: Evolution Problems I. Springer-Verlag: Berlin, 1992.
E. Dean and P. Gubernatis, “Pointwise control of Burgers equation—a numerical approach, ” Computer and Mathematics, vol. 22, pp. 93–100, 1991.
M.C. Desai and K. Ito, “Optimal controls of Navier-Stokes equations, ” SIAM J. Control and Optimization, vol. 32, no. 5, pp. 1428–1446, 1994.
M. Hinze and S. Volkwein, Analysis of instantaneous control for the Burgers equation, Submitted, 1999.
E. Hopf, “The partial differential Equation u t + uu x = μu xx, ” Commun. Pure Applied Mathematics, pp. 201–230, 1950.
Y. Ikeda, “Optimal active control for Burgers equation, ” Annual Research Briefs 1992, pp. 219–224, 1994. Center for Turbulence Research.
K. Ito and S. Kang, “A dissipative feedback control synthesis for systems arising in fluid dynamics, ” SIAM J. Control and Optimization, vol. 32, no. 3, pp. 831–854, May 1994.
K. Ito and K. Kunisch, “Augmented Lagrangian-SQP-methods in Hilbert spaces and application to control in the coefficient problems, ” SIAM J. on Optimization, vol. 6, no. 1, pp. 96–125, 1996.
S. Kang, “A control problem for Burgers Equation, ” PhD thesis, Department Mathematics of Virginia Polytechnic Institute and State University, April 1990.
S. Kang, K. Ito, and J.A. Burns, “Unbounded observation and boundary control problems for burgers equation, ” in 30th IEEE Conf. on Decision and Control, Dec. 1991, pp. 2687–2692.
A. Kauffmann, “Optimal Control of the Solid Fuel Ignition Model, ” PhD thesis, Department of Mathematics, Technical University of Berlin, Oct. 1998.
H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, AcademicPress: London, 1989.
E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1978.
K. Kunisch and S. Volkwein, “Control of Burgers' equation by a reduced order approach using proper orthogonal decomposition, ” JOTA, no. 2, pp. 345–371, 1999.
D.G. Luenberger, Optimization by Vector Space Methods, John Wiley & Sons, Inc., 1969.
H.V. Ly, K.D. Mease, and E.S. Titi, “Some remarks on distributed and boundary control of the viscous Burgers equation, ” Preprint, Feb. 1996.
H. Maurer and J. Zowe, “First and second order necessary and sufficient optimality conditions for infinitedimensional programming problems, ” Math. Programming, vol. 16, pp. 98–110, 1979.
A. Pazy, Semigoups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag: New York, 1983.
J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag: Berlin, 1983.
R. Temam, Navier-Stokes Equations, Studies in Mathematics and its Applications. North-Holland: Amsterdam, 1979.
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, volume 68 of Applied Mathematical Sciences, Springer-Verlag: New York, 1988.
S. Volkwein, “Mesh-independence for an augmented Lagrangian-SQP method in Hilbert spaces, ” SIAM J. Control and Optimization, vol. 38, no. 3, pp. 767–785.
S. Volkwein, “Application of the augmented Lagrangian-SQP method to optimal control problems for the stationary Burgers equation, ” Computational Optimization and Applications, vol. 16, pp. 57–81, 2000.
S. Weerakoon, “An Initial Value Control Problem for the Burgers' Equation, ” PhD thesis, Department of Mathematics of Pennsylvania State University, Dec. 1984.
A. Wouk, A Course of Applied Functional Analysis, Wiley-Interscience: New York, 1979.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Volkwein, S. Distributed Control Problems for the Burgers Equation. Computational Optimization and Applications 18, 115–140 (2001). https://doi.org/10.1023/A:1008770404256
Issue Date:
DOI: https://doi.org/10.1023/A:1008770404256